Gao, Weidong; Geroldinger, Alfred On products of \(k\) atoms. (English) Zbl 1184.20051 Monatsh. Math. 156, No. 2, 141-157 (2009). Let \(R\) be a Noetherian integral domain and denote by \(V_k(R)\) the set of all integers \(m\) with the property that there exists an element of \(R\) having factorization into irreducibles of length \(k\) and \(m\). The authors show that for a large class of domains the set \(V_k(R)\) is very close to an arithmetic progression. They work actually in a more general setting, dealing with factorizations in atomic monoids, i.e. commutative cancellative semigroups with unit element in which every non-unit is a product of irreducible elements. Reviewer: Władysław Narkiewicz (Wrocław) Cited in 2 ReviewsCited in 37 Documents MSC: 20M14 Commutative semigroups 13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) 11R27 Units and factorization 13A05 Divisibility and factorizations in commutative rings 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations Keywords:non-unique factorizations; sets of lengths of factorizations; Krull monoids; atomic monoids; commutative cancellative semigroups; products of irreducible elements PDFBibTeX XMLCite \textit{W. Gao} and \textit{A. Geroldinger}, Monatsh. Math. 156, No. 2, 141--157 (2009; Zbl 1184.20051) Full Text: DOI References: [2] Baginski P, Chapman ST, Hine N, Paixao J (2008) On the asymptotic behavior of unions of sets of lengths in atomic monoids. Involve, to appear [9] Foroutan A, Geroldinger A (2005) Monotone chains of factorizations in C-monoids. In: Chapman ST (ed) Arithmetical Properties of Commutative Rings and Monoids, Lect Notes Pure Appl Math 241, pp 99–113. Boca Raton, FL: Chapman & Hall/CRC · Zbl 1095.20040 [10] Freeze M, Geroldinger A (2008) Unions of sets of lengths. Funct Approximotio, Comment Math, to appear · Zbl 1228.20046 [22] Schmid WA (2007) A realization theorem for sets of lengths. Manuscript This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.